Rigging for Popup Book
I found a way to do this with a simple rig (see attached blend file).

There are two armatures, one for the book, the other for the popup.
The book armature (yellow bones) above controls opening/closing the book:

Each page (L and R) has two bones; the innermost bone controls opening/closing via simple rotation. The outermost ones are there only for targeting the IK constraints in the popup armature.
The tails of the inner bones (Page_L and Page_R) are positioned so they are attachment points of the popup.
A rotation constraint on Page_L and Page_R makes the book open on axis for the book binding.
Rotate Page_L to open/close the book.
It only needs forward kinetics for opening/closing -- no IK constraints.

This image shows the Popup armature (orange bones):


The Popup_Center bone is the root bone; its head is where the popup middle is attached to the book and its head is where the popup apex is attached.
The Popup_L and Popup_R bones extend to the book attachment points.
IK constraints keep the popup attached to the book pages:

Popup_L IK constraint targets Page.001_L of the Book armature
Popup_R IK constraint targets Page.001_R



By rotating Page_L to close the book, Popup_L is pushed, due to the IK constraint.
Popup_L pushes on Popup_Center, forcing it to rotate towards the book.
Popup_R also pushes on Popup_Center due to its IK constraint.
Some additional notes:

I probably could have gotten away with using only a single armature. With two armatures, I could modularize multiple popups.
A control bone for managing the rotation of Page_L would have been nice, but I don't know how to do that.
I probably could have been more flexible with the bone placements, using offsets, but I haven't learned yet how to do that.
Blender is amazing. What I thought would be difficult or impossible is actually easy, once I learned how IK targets work.
I would have liked to make an animated gif of how this works in a 3d viewport, but I don't know how do that (advice, anyone?)

Here is an updated blend file: